# Find the coordinates of the points which divide

Question:

Find the coordinates of the points which divide the line segment joining the points (−4, 0) and (0, 6) in four equal parts.

Solution:

The co-ordinates of the midpoint $\left(x_{m}, y_{m}\right)$ between two points $\left(x_{1}, y_{1}\right)$ and $\left(x_{2}, y_{2}\right)$ is given by,

$\left(x_{w}, y_{m}\right)=\left(\left(\frac{x_{1}+x_{2}}{2}\right),\left(\frac{y_{1}+y_{2}}{2}\right)\right)$

Here we are supposed to find the points which divide the line joining A(−4,0) and B(0,6) into 4 equal parts.

We shall first find the midpoint M(x, y) of these two points since this point will divide the line into two equal parts

$\left(x_{m}, y_{n}\right)=\left(\left(\frac{-4+0}{2}\right),\left(\frac{0+6}{2}\right)\right)$

$\left(x_{m}, y_{m}\right)=(-2,3)$

So the point M(−2,3) splits this line into two equal parts.

Now, we need to find the midpoint of A(−4,0) and M(−2,3) separately and the midpoint of B(0,6) and M(−2,3). These two points along with M(−2,3) split the line joining the original two points into four equal parts.

Let  be the midpoint of A(−4,0) and M(−2,3).

$(e, d)=\left(\left(\frac{-4-2}{2}\right),\left(\frac{0+3}{2}\right)\right)$

$(e, d)=\left(-3, \frac{3}{2}\right)$

Now let  bet the midpoint of B(0,6) and M(−2,3).

$(g, h)=\left(\left(\frac{0-2}{2}\right),\left(\frac{6+3}{2}\right)\right)$

$(g, h)=\left(-1, \frac{9}{2}\right)$

Hence the co-ordinates of the points which divide the line joining the two given points are $\left(-3, \frac{3}{2}\right),(-2,3)$ and $\left.\left(-1, \frac{9}{2}\right)\right)$.